# Resolved Problems of Linear Equations

• Introduction

• We remember that...

• 20 Resolved Problems of Linear Equations

## Introduction

A linear equation is a polynomial equation whose degree is 1, because the highest degree of the monomials is one (the highest power is 1, in x). Because of this, we'll have one solution, if it has a solution, some don't.

In this section we are going to resolve exercises with linear equations, the typical type of equations we'll encounter in out day to day life. The solution requires a lay out and a resolution of the equation with a unknown factor.

The exercises are in order of increasing difficulty, starting with simple notions like representing the double, triple and consecutive numbers with algebra.

Although we suppose we know how to resolve this type of equations,

## We remember that...

• If we obtain an impossible equality, there isn't a solution. For example, if we obtain 1 = 0. This happens for example with the equation x = x + 1, which is like saying a number is equal to its consecutive, which is false. Then, it's logical that the equation doesn't have a solution.

• If we obtain an equality that always happens, either value is a solution, and the solution is all the real numbers. For example, if we obtain 0 = 0. This happens with the equation x = x, which is the same as saying a number is equal to itself, which is always true.

If we still don't master linear equations (with fractions, parenthesis, joint parenthesis...) we can find resolved exercises in this section.

# Resolved Problems step by step

Problem 1

Write the following expressions in algebraic form:

1. The double of a number x.

2. The triple of a number x.

3. The double of a number x plus 5.

4. The triple of a number x squared.

5. The three fourths parts of a number x.

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Problem 2

Find the numbers, in each case, that apply:

1. Double it plus 5 is 35

2. Adding its consecutive number we obtain 51

3. Adding its double, its half and 15 we obtain 99

4. Its quarter part is 15

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Problem 3

Marta is 15 years old, and that's a third part of her mother’s age. How old is her mother?

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Problem 4

How much does a rope measure if the three quarters of it measures 200 metres?

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Problem 5

Obtain three consecutive numbers that added up gives 219.

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Problem 6

We travel a 1 km long road at 6km/h. How long will it take us to get to our destination?

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Problem 7

James puts 25 dollars in his safe, which is a fourth of the money in the safe to begin with. How much was in it to start with?

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Problem 8

Anne's father is 5 years younger than her mother, and half her mother's age is 23. How old is Anne's father?

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Problem 9

Stefanie is 16 years old and her two brothers are 2 and 3 years old. How many years need to go by so that the double of the sum of her two brothers’ ages is the same as her age?

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Problem 10

Given a number, adding half of it, double it and triple it is 55. What number is it?

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Problem 11

Vincent spends 20 dollars on a trousers and a shirt. He doesn't know what each item costs, but he does know that the shirt is worth two fifths of the trousers. How much do the trousers cost?

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Problem 12

The difference between two numbers is 17, and the double of the smallest number is 26. Which numbers are they?

And if 26 is double the biggest number, what numbers are they?

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Problem 13

5 years ago, Mike was three times the age of his cousin Joe, who is 15 years old. How many years need to go by for Joe to have Mike's current age?

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Problem 14

We have 3 fish tanks and 56 fishes. The sizes of the fish tanks are small, medium and large. The small one is half the medium one and the big one is double the medium.

We don't have a preference in how the fish are distributed but we decide that in each of them there will be a proportional amount of fish for the size.

How many fish will we put in each tank?

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Problem 15

We want to distribute 510 sweets amongst 3 kids, in a way that two of them have half of the sweets, but one of these two has half of the others. How many sweets will each child have?

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Problem 16

One third part of the spoons of the house was in the dishwasher and the rest in the drawer. But half the spoons in the drawer, 15, are going to the table. How many spoons are there in the dishwasher?

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Problem 17

In a shop they sell a third of their products in two days. The next day they receive from a warehouse half of the quantity they sold, which is 15 units. How many units did they sell in the first two days? How many units are there in the shop after they sell?

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Problem 18

Mark has 400 dollars and Rose has 350. Both of them buy the same book. After the purchase, Rose has five sixths of the money that Mark has left. Work out the price of the book.

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Problem 19

Esther has three times the money Penny has and half of what John has. John gives Penny and Esther 25 dollars each. Now, Esther has the same amount as John. How much did they have to start with? And afterwards?

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Problem 20

In a house, the water deposit is at 2/7 its capacity. Three people have showers: The first one uses one fifth of the quantity in the tank; the second uses a third of what's in the tank and the third, three fourths of the firsts' quantity. Which is the capacity of the tank and how much water did the first two use is we know that the third used 10 litres to shower?

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